#453496

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,16,13,15,19,14,29)(11,20,30,22,24,36,23,17)(12,18,27,40,28,34,31,26)(21,35,37,32,25,38,39,33);(3,7)(5,8)(9,11,14,23,15,24,16,30)(10,17,13,20,19,22,29,36)(12,25,27,39,28,21,31,37)(18,33,26,38,34,32,40,35);(1,3,4,7)(2,5,6,8)(9,12,15,28)(10,18,19,34)(11,21,24,25)(13,26,29,40)(14,27,16,31)(17,32,22,33)(20,35,36,38)(23,37,30,39);(9,13,16,19,15,29,14,10)(11,22,30,36,24,17,23,20)(12,26,31,34,28,40,27,18)(21,33,39,38,25,32,37,35)

Polycyclic presentation:

2^9|a^2=efgh,b^2=e,c^2=g,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=dei,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360692);